Spherical calculator



May 2, 1950 J. M. THOMAS SPHERICAL CALCULATOR May 2, 195o J. M. THOMAS2,506,251

SPHERICAL CALCULATOR Filed Feb. 26, 1945 2 Sheets-Sheet 2 Patented May2, 1950 UNITED STATES PATENT Ormes SPHERICAL CALCULATOR Joseph MillerThomas, Durham, N. C.

Application February 26, 1945, Serial No. 579,840

3 Claims.

This invention relates to calculating devices and more particularly toan apparatus for use in celestial navigation for determining theposition of a ship, airplane or the like.

More speciiically this invention has as its object to provideaninstrument capable of calculating directly the latitude and longitudeof a given point on the terrestrial sphere from the Greenwich hourangle, declination and altitude of two points on the celestial sphereobserved simultaneously from the given point.

For the great circle track between two points whose latitude andlongitude are given the instrument of this invention can be used tocalculate directly the spherical distance in miles and the latitude andlongitude at any distance along the track and can be adapted tocalculate the azimuth of a point on the celestial sphere, the

course at any distance on the great circlev track,

or the unknown parts of any spherical triangle. Other objects of thisinvention will appear from the following description and drawingsillustrating this invention wherein:

Figure 1 is-a diagrammatic construction illustrating the basicprinciples of this invention.

Figure 2 is a plan View of an embodiment of a calculating deviceaccording to this invention. Figure 3 is a sectional view taken on line3-3 f of Figure 2.

Figure 4 is an elevational View showing the appearance of the securingpins used in the calculating device of Figure 2.

Figure 5 is a cross-sectional view taken on line 5 5 of Figure 3.

Figure 6 is a partial schematic view showing the declination circlediagram obtained from the values of declination of two stars used indetermining the latitude and longitude from the device of thisinvention.

Figure '7 is a partial schematic View similar to Figure 6 but showingthe altitude circle diagram.

Figure 8 is a partial schematic view showing the circle diagramcorresponding to settings of the GHA arms for the stars of Figures 6 and'7.

Figure 9 is a schematic view representing the final closure of thelinkage formed by the various arms whereby latitude and longitude can beread off the instrument.

Figure 10 is a vertical sectional detail view taken radially through thesecond GHA Vernier of the device of Figure 2.

Referring to Figure 1, a geometrical construction is disclosed whereinthe paper represents the plane of the meridian viewed from the east.

The meridian is represented by the circle with center at O defined bypoints A, D, F', N, C', S, F, C and B. CC is the diameter oi a starsdiurnal circle which' has been rotated through 90 into the plane of thevpaper. The rotated position of the diurnal circle is the circle CHC. Thepoint I-l is the rotated position of the star. Arc CH is therefore thelocal hour angle. The circle FJF' represents the stars altitude circlerotated 90 into the plane of the paper around the diameter FF". Arc FJis therefore the azimuth. If the circles were rotated back into theiroriginal positions on the celestial sphere, l-I and J would coincidesince J is the rotated position of the star on the altitude circle. Thiscoincidence would occurV in the perpendicular to the paper erected at G.GH is therefore perpendicular 'to CC'. If GH is prolonged and 0D drawnparallel to KH, where K is the center of CC', a point E will bedetermined as the intersection of GH and OD. Since OE is equal to KH(KHEO being a parallelogram), OE is hence equal to KC, which like KH, isa radius of circle CHC'.

Reference is ymade to the accompanying Figures 2 and 3. rThe instrumentproper consists Voi a hollow cylindrical'base I, in which rotate fourother cylinders 2, 3, 4 and 5. Cylinder 2 carries radial arm 6 rigidlyattached. Cylinder fi carries: chordal armsl and S which have groovedends 9 and Ill sliding in tracks I3 and Iii; chordal arms I5 and I6which have grooved ends l1, I8, i9 and 20 sliding in tracks 2! and 22,and which are set by clamp devices 23 and 2t; and aligner 25 rigidlyattached. Cylinder 5 carries: chordal arms 2t and 27 which have flangedends 28, 29, 30 and 3| sliding in tracks 32 and 33 and which are set byclamp devices 34 and 35; and aligner 36 rigidly attached. In the zeroposition of the cylinder, aligner 25 is at right angles to tracks 32 andS3 and extends parallel to arms 26 and E?.

Base I has a groove 3'I in which slides carriage .se set by clamp as.Arm e is similarly equipped with carriagedo and clamp 43 thereon. Armsl', 8, I5, i 6, 26 and 2 are pierced by slots in which move carriagesv,without fastening devices. Arms 7 and 8 have two carriages each.

Each carriage is pierced by a vertical cylindrical hole. Four pins withcollars, two-like that shown at 4I in Figure 4 and two like that' shownat 42 in Figure 4, are provided. A pin 4I is used rto align the carriage"of arm 2t (or 27) over aligner 25; or to align the carriage of arm I5(or I6) under aligner 36; or to align the carriage vof arm 26 (or 21)over the carriage of arm I5 (or I6) over a .carriage of armf'l (ort).Apin 3 42 is used to align a carriage of 1 (or 8) over carriage 35 (or40).

Clamp devices 43 and 44 lock 4 to 3 and 5 to 4 respectively and alsoserve as knobs for turning the cylinders.

The upper face of cylinder l carries a Vernier 45 which is attached byscrews 46 and 4l and which projects over cylinder 2. The upper face ofcylinder 2 carries a raised Vernier 48. The outer edge of theupperiaceof cylinderi carries the GI-IA scale graduated clockwise and the inneredge carries the longitude scale graduated clockwise for west andcounterclockwise for east. The outer edge of the upper face ofrcyliniier @i4 carries the longitude pointer and double .Vernier 49; theinner edge carries the latitude `scale graduated clockwise for northYia'ndzthedistance pointer and Vernier 50. The outer edge of the upperface of cylinder 5 carries the "latitude pointer and double Vernier 5land the distance scale. `v(If'the :distance @Vernier @and f-sc'ale :arefrotated through thefsame: angle with respect ito the cthermarkings,thezerdfoithe distance scale can bezput inca, different position.)

.--and clampfdevicer53 locks cylinderil tocylinderl.

-10nz the base al Aisffdrawvnfundamental .circle'l'iL .as .shown in'Figure f2. lIts size determines'the faccuracy. If this4 circlelis:(iveiinches .in 'diameter and :theinstrumentIismanufactured with cthe;same :.precsion; as: :a good transit the instrumental ierrorzshould :notexceed one minute.

. described here, ithei carriages' 33,-* and 4t are permanently .marked"1' wand tf2 respectively; :and the carriages fon 1.5 :and l5 arelikewise .accordingly the 'estar v`with -larger declination.Thescarriages of 1, 5.8, 2.6: and;2'l maybe marked "1. or 2 .as required'for sa @particular i calcula- '.tion.

"The .end portions offlthe' various guideslotsror grooves are.provided-with enlarged' openl'portions permitting the removal :orTinsertion fof the `car- 'Such enlarged portions lare "Therezis :aminimum. difference 1 of ideclination,

foffGI-IA 'andof Valtitude.below Whichtheiinstruv'Several modiiications:and extensions 'of i.this

theory .were necessary in lorder todevise the-pres ent instrument:

(il) The constructionfmust'lie done initerms of `v coordinates ratherthan 'elementsfof an Euler `triangle. vmo'dication, :although Teasy tomarke, -is 'of :the utmost importance Tfrom fthe standpoint ofconvenience.

'(2) The fc'ireles employed Imust -"all i be `concentric.

G3) 'The .solution o'f tthei'fundamental yposition isf-equivalent tothesimultaneous-solutionfof' three 'lspl'ierical 'triangles.

lit' is ytodae understoo'dlthat': in the faccompany- Aing drawings Nyand vcoordinates are to *be measured clock-wise and thatN--as-alzvpliedfto:altitude meansl ahovsthe horizon. 'InFigure'lthe :circleof unit radius--Withfcenter ati' 0 isl now considered. FromanypointsAoniits circumference 4 lay off arcs AN=plus 90, AS=minus 90.Con struct BA equal to the latitude of a point P, BC equal to thedeclination of a star, SF equal to the altitude of the star as seen fromP, arc 5 BD equal to the local hour angle at P, and OE which passesthrough D and equals one half chord CC (that is, equals the lcosine ofthe declination) It is then easy to show from the preceding theoreticaldiscussion that chords CC,.F.".E"" and the l0 perpendicular from E-'toCC'iare-concurrent in a point G.

If the latitude and longitude of P are not given, from the coordinatesof two stars we can iconstructthe three partial diagrams shown in isFigures 6, 7 and 8. The solution of the funda- `'mentalproblem consistsin rotating the declina- -.i:ien-and altitude diagrams until theintersections of corresponding chords coincide with the feet fofitheperpendiculars from Ei, E2 to the declina- 20 tion chords. This isaccomplished mechanically vas follows: the'carr'iages" 'nandionvthe GHAgroove el. andarm .G are .fastened :at theiproperdistancesiromithefcenter ilatEifEz, aswillpresently be described,thelGI-IA gro'ovezlan'd larm 'l are'setatrthe :'appropriatevangles,rlas'will be presfently shown, .an'dfthe three: cylinders 2- andl 3 'are lockediso as tcfrotate together (Figur-eid). One "ofV theAcarriages on .1 isfpinned-'to lthefcaririage atlEi (or 'Einaudi onetofthecarriages on 8 :iois pinnedtothercarriageuatEziorED. thedeclinationfcirc zFigure-) rotates Iwith'f respect tothe' GHA-'circl'es(iFig-.ure`8), armslvand are l constrained byitheir tcarriagesA t0continue -to-'pa'ss through Ei .and E2 and by-.Ltheirfgroovedfendsto '35remain .perpendicL-ilarfl to :the declination rarms l 5 and it. .Thelast mentionedfrare..'setfin-the:apz'propriate. position, '.-a's .willfbe ishown, with-"respect to the iiindarnental f'circle (miade .in these-later drawings "to "have fthe -isame radius ias fthe inner l-ledge `ofcylinderi4) :asfshownLinFigure 6. I'The :altitude .arms-ZScandZl-aaresetpaswill be subsef quentlydescribed.' atb-'theirappropriatei'placeseas shown in Figure?. "The#sec'ond-'carragevr on 4arm'l '(say) is'fpi'nned at lltotirez-intersectionLofthe 45 star N011altitude arm :strand eternal decima- .tionarm llo'y meansdf thezslidingcarriages on those warms. `The =system Ils theniirotate'd until :theintersection ofthe stari'No. T2 'altitudelarm 21 and star No. 2declination arm l5 -co'mesrover o. armi 8 4at2". The mechanismcan-then'belocked 'in `position 'by pinning'ithrcu'gh three `verticallyaligned carriages. Figuref show'sfvthe'nal positi'on withpins at l'and2.

It is clear from Figure 1 thatthela'titude isthe 55-displaoement ofthe`zenith 1'Afrom the point B. t can be'readfonf-the .declination scale bymeans of thelatitudepointer iixed'ia-tfA. Similarlyfit Vis cle-arithatthe displacement of' B from'the'point D with GI-IA equal t0 zeroisithelongitude and G0 .f can be read ontheflongitudescale by thelongitude pointer ixedratB.

There are, of course, in general two solutions whichmu'st'"be-'distinguished ("say, by vrougi-ily ob- 'served azimuth). y a'Theffollowing specc'step's showhowfto'operate i the. calculator:

D. `Release 'allclamps vl. 'Set vpointer 5l opposite-declination 'oflstar No. l. Set clamp 44.

l'2. PinsocketfSS'to the carriage onf'armf l5 land to onecarriage'on'azrm "Set'fcla-mpZS.

3. Align GI-IA pointer No. lwith'Ocn'cylinder 3 Land with pointer 249."Set clamps 213,53. v'74."Pin carriage 38 tothe fsec'ond-*carriage'on f5 I`"arm f1. Set'clamp`39.

I"5. Repeat the above for star No. 2. Speciiically:

5.0. Remove pin from 36 and from the carriages on 1 and I5. Releaseclamps 43, 44, 53.

5.1. Set pointer 5I opposite declination of star No. 2. Set clamp 44.

5.2. Pin socket 36 to carriage on arm I6 and to one carriage on arm 8.Set clamp 24.

5.3. Align GHA pointer No. 2 with 0 on cylinder 3 and with pointer 49.Set clamps 43, 52, 53.

5.4. Pin carriage 40 to the second carriage on 'arm 8. Set clamp 46.

`5.5. Remove pin from 36 and from the carriages :on 8 and I6. Releaseclamps 43, 44, 52, and 53.

' 6. Set pointer 5I opposite altitude of star No. 1. Clamp 44.

7. Pin socket 25 to the carriage on arm 26. Clamp 34.

'7.1.l Remove pin from 25 and from the carriage on 26. Release clamp 44.

8. Set the altitude of star No. 2 as follows:

8.1. Set pointer 5| at the altitude of star No. 2. Clamp 44.

8.2. Pin socket 25 to the carriage on arm 21. Clamp 35.

8.3. Remove pin from 25 and from the carriage on 21. Release clamp 44.

9. Set the GHA pointer No. 1 opposite the GHA of star No. 1 and the GHApointer No, 2 opposite the GHA of star No. 2. Clamp 52, 53.

10. Bring the arm 26 over the intersection of arms 1 and I5. Alignvertically the carriage on arm 26, the unpinned carriage on arm 1, andthe carriage on arm I5. Pin the three carriages together.

11. Bring the arm 21 over the intersection of arms 8 and I6. Alignvertically the carriage on arm 21, the unpinned carriage on arm 8 andthe carriage on arm I6. Pin the three carriages together.

12. Read the latitude by Vernier 5I and the longitude by Vernier 49.

For the sake of deiiniteness, it has been lassumed that star No. 1 hasthe algebraically smaller altitude, and the algebraically smallerdeclination. It is always possible to designate as No. 1 the star withsmaller declination. If, however, the star designated as No. 2 has thesmaller altitude, the roles of arms 26 and 21 must be interchanged. Inother words, arm 1 is always used for the star with the algebraicallysmaller declination and arm 26 for the star with the algebraicallysmaller altitude.

Steps 1-2 fasten arm I5 (No. 1 star) to cylinder 4, and steps 5.15.2fasten arm I6 (No. 2 star) to cylinder 4, realizing the situation shownin Figure 6.

Steps 3-4 fasten carriage 38 to cylinder I at the radial position E1 inFigure 8. Steps 5.3-5.4 fasten carriage 40 to cylinder 2 at the radialposition Ez in Figure 8.

Steps 3-4 at the same time constrain arm 1, which is alwaysperpendicular to arm I5, henceforth to pass over Ei. Steps 5.3-54 at thesame time constrain arm 8, which is always perpendicular to arm I8,henceforth to pass over E2.

Steps 6-7 fasten arm 2B (No. 1 star) to cylinder 5, and steps 8.1-8.2fasten arm 21 (No. 2 star) to cylinder 5, realizing the situation shownin Figure 7.

Step 9 sets and clamps cylinders I, 2 and 3 in their iinal relativepositions shown in Figure 8.

Henceforth there are essentially just the three cylinders shown inFigures 6, 7 and 8. If the unit composd of cylinders I, 2 and 3 isregarded 'as iixed, during steps 10 and 11 cylinders 4 and 5 rotate withrespect to that unit. It is unnecessary to specify how the rotationtakes place. The theory developed in connection with Figures 1 and 9shows that the motion is possible. If the parts of the instrument areaccurately machined, lthe cylinders will follow the motion of the arms26 and 21 without difculty.

Step 10 makes the lines representing arms 1,

I5 and 26 pass through a point and continue to do so. The point ofintersection moves, in step 11, until it reaches the nal position I inFigure 9. f Step 11 makes the lines representing arms 8, I6 and 21 passthrough 2 in Figure 9 and locks the Whole mechanism by pinning through'the three carriages.

While certain speciiic embodiments of this invention have been shown anddescribed herein it will be understood that various additionalmodifications may be made within the spirit of the invention. Thereforeno limitations upon the invention are intended other than are imposed bythe scope of the appended claims.

I claim:

1. In a calculator for solving spherical triangles wherein thecoordinates of two points of the triangle are known and it is desired todetermine the coordinates of the third point, a plurality of nestedmembers means constraining said nested members for mutual rotation abouta common axis, said nested members being provided with measuring scales,a free linkage system having elements which are adjustable in length anddirection and are operatively connected to the nested members for suchadjustment, means for adjusting the linkage elements in accordance withthe coordinates of the two known points oi the triangle as measured onsaid scales, a free carriage slidably mounted on one linkage element,and means for rotating the nested members to an indicating position atwhich said free carriage is in registry with the intersection of twoother linkage elements whereby the linkage may be locked by pinningthrough at the point of registry.

2. In a calculator, a cylindrical outer cupshaped member, a radialgroove in the inner bottom surface of said cup-shaped member, a secondcylindrical cup-shaped member rotatably iitting within the outer member,a central aperture in the bottom wall of said second member, a radialarm carried by said bottom wall in said aperture, a carriage slidablytting said groove for radial adjustment thereon, another carriageslidably tting said radial arm for radial adjustment thereon, a thirdcylindrical cup-shaped member rotatably iitting within said secondmember, a rectangular central aperture in said third member, a pair ofhorizontal guide tracks formed in a rst two opposing walls of saidrectangular aperture, a first pair of arms slidably tting said guidetracks and bridging said first two opposing walls, means constrainingsaid first pair of arms to perpendicular positions with respect to saidfirst two opposing walls, a pair of horizontal guide tracks formed inthe other two opposing walls of the rectangular aperture, another pairof arms slidably tting the horizontal guide tracks in said other twoopposing walls, means constraining said another pair of arms toperpendicular positions with respect to said other two opposing walls, acylindrical disc element rotatably fitting within said third cup-shapedmember, a rectangular central opening formed in said disc element, apair of horizontal guide tracks formed in two opposing walls of saidopening, a pair of bridging elements sidablyzfttingime ftr asksin-theropposingjfwals :of -saidf opening; meansconstran-ing the bridging:elementsV tosperpendicularf positions with respect to` lthe :sa-idopposing-v Wa-llszof thefopening, means 4for pivotal'lylock-ing,oneofsaid'rstpair of arms `tri-:true cerriagevnsad radaigroove,means for `pivotaliy locking thenther'of sal-d rst-pair of arms to thecarriage onsa'dradal arm, means for locking one ofthefbrdging-eiernents; oneof the armsein said .other ltwo fopposngwall-sandone'of thefarmsinsad rst two opposing Walls insuperimposed-relationship, and ymeans -for locking the other of thebridgingelements, the otiner'offthe armrs'in said other two opposingWalls'and the Vother ofthe arms v-in-sad .ti-rst two opposing-Walls insuperimposed relationship.

3. The structure of claim-2, andwheren the rim of the third cup-shapedmember projects .above theY leve of. the rimvof the second cup- `shaped.member, and wherein a. ringmemberis .provided,..sad ringinember beingformed to rotativelyt around ,the vouter `perip'heryrof the: .rim

bear on the top of the rim of the second cupe shaped;- member.

JOSEPH lMlrLElR, THOMAS REFERENCES :CITED The following referenceslarefof record inthe le of this.. patent:

UNITED STATES l:"-ATE1\I"JI'S.`

Number Name` Date 1,770,461 Cogshall 15,' `1930 2,105,103 Swtzin-WhiteaJam 11,.. 1938 213%;981. Chamberlin" Jne'Z', 19.45 2,421,965 SagBereira.June10, 1947 FOREIGN PA'rEms- 4 Al\unf1ber Country Date;

946 GreatBritan Apr; 3; ,1866 543,886 Great Britain Mar. 18, `1942

